8R: Pythagorean Theorem, -...
Transcript of 8R: Pythagorean Theorem, -...
Grade 8 Review # 9 (Pythagorean Theorem, Distance, Midpoint)
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Ahrens (2014)
Label a Right Triangle
Legs
Opposite the right angle Longest of the 3 sides
2 sides that form the right angle
Labels for a right triangle
ca
b
Hypotenuse
8R: Pythagorean Theorem, Distance, and Midpoints
formula
In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse ( c).
a2 + b2 = c2
Find the Hypotenuse
4 in
7 in
a2 + b 2 = c 2
42 + 7 2 = c 2
16 + 49 = c 2
65 = c 2
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root &Label Answer
Find a missing leg
a2 + b2 = c2
52 + b2 = 152
25 + b2 = 225
25 25
b2 = 200
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer5 ft
15 ft
Pythagorean Triplets
3
4
There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triplets.
345 is the most famous of the triplets. If you recognize the sides of the triangle as being a triplet (or multiple of one), you won't need a calculator!
5
Pythagorean Triplets
Pythagorean Triples
3 4 55 12 137 24 258 15 17
Multiples of these combinations work too!
Pythagorean Theorem Examples
Grade 8 Review # 9 (Pythagorean Theorem, Distance, Midpoint)
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Ahrens (2014)
Corollary (Converse) to the Pythagorean Theorem
Corollary to the Pythagorean TheoremIf a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 = a2 + b2, then the triangle is a right triangle.
If c2 ≠ a2 + b2, then the triangle is not a right triangle.
b = 4 ft
c = 5 fta = 3 ft
Corollary (Converse) to the Pythagorean Theorem
Corollary to the Pythagorean Theorem
In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true.
Test the Pythagorean Theorem. If the final equation is true, then the triangle is right. If the final equation is false, then the triangle is not right.
Oct 289:28 AM
Is it a Right Triangle?
Write Equation
Plug in numbers
Square numbers
Simplify both sides
Are they equal?
8 in, 17 in, 15 in
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289
Yes!
Corollary Example
Distance of a Vertical Line
If you have two points on a graph, such as (5,2) and (5,6), you can find the distance between them by simply counting units on the graph, since they lie in a vertical line.
The distance between these two points is 4.
The top point is 4 above the lower point.
Distance of a Slanted Line
Most sets of points do not lie in a vertical or horizontal line. For example:
Counting the units between these two points is impossible. So mathematicians have developed a formula using the Pythagorean theorem to find the distance between two points.
Grade 8 Review # 9 (Pythagorean Theorem, Distance, Midpoint)
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Ahrens (2014)
FInd the Distance
Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red.
c2 = a2 + b2
c2 = 32 + 42c2 = 9 + 16c2 = 25c = 5a
bc
The distance between the two points (2,2) and (5,6) is 5 units.
formula derivation
Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Formula.
(x1, y1)
length = x2 x1
length = y2 y1
d
c2 = a2 + b2
d2 = (x2 x1)2 + (y2 y1)2
d = (x2 x1)2 + (y2 y1)2This is the distance formula now substitute in values.
d = (5 2)2 + (6 2)2
d = (3)2 + (4)2
d = 9 + 16
d = 25
d = 5
(x2, y2)
Distance Formula
Distance Formula
d = (x2 x1)2 + (y2 y1)2
You can find the distance d between any two points (x1, y1) and (x2, y2) using the formula below.
how far between the xcoord. how far between the ycoord.
Distance Example
When only given the two points, use the formula.
Find the distance between:Point 1 (4, 7)Point 2 (5, 2)
for formula
d = (x2 x1)2 + (y2 y1)2
Distance Geometry Problem
You can use the Distance Formula to solve geometry problems.
A (0,1)B (8,0)
C (9,4)D (3,3)
Find the perimeter of ABCD.Use the distance formula to find all four of the side lengths.Then add then together.
BC =BC =
CD =CD =
AB =AB =
DA =DA =
Distance Using Pythagorean Theorem instead of Distance Formula
Grade 8 Review # 9 (Pythagorean Theorem, Distance, Midpoint)
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Ahrens (2014)
Midpoint Formula
(3, 4) (9, 4)
Find the midpoint of the line segment.
What are the coordinates of the midpoint?How is it related to the coordinates of the endpoints?
Midpoint = (6, 4)
It is in the middle of the segment.
Average of xcoordinates.Average of ycoordinates.
Midpoint Formula
The Midpoint Formula
To calculate the midpoint of a line segment with endpoints (x1,y1) and (x2,y2) use the formula:
(x1 + x2 y1 + y222
, )
The x and y coordinates of the midpoint are the averages of the x and y coordinates of the endpoints, respectively.
Midpoint Example
The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)Use the midpoint formula:
Substitute in values:
( x1 + x2 y1 + y222
, )2 + 8 , 5 + 12 2( )
Simplify the numerators:10 62 2
,
Write fractions in simplest form:
( )(5,3) is the midpoint of AB
M
Midpoint Example
If point M is the midpoint between the points P and Q. Find the coordinates of the missing point.
Use the midpoint formula and solve for the unknown.
M (8,1)
P (8,6)
Q = ? (x2,y2)
(x1 + x2 y1 + y222
, )Substitute
Multiply both sides by 2
Add or subtract(8, 8)